## Description

This textbook is meant for students of first course on Differential Calculus. The book provides balanced coverage of both theory and practice of concepts. A wide variety of problems and lucid language will help students understand and apply the concepts easily. The step-by-step solutions to solved problems enables clear understanding. A variety of solved and unsolved problems have been incorporated to help students ace university and competitive examinations.

## Table of Content

**Chapter 1 Successive Differentiation **

1.1 Introduction

1.2 nth Derivatives of Some Standard Forms of Functions

1.2.1 When y = eax then Dy =aneax

1.2.2 When y = (ax + b)m, where m ≥ n, then

1.2.3 When y = sin(ax+b), then Dny = an sin(n p/2 + ax + b)

1.2.4 When y = cos (ax+b), then Dn y = an cos (n p/2 + ax + b)

1.2.5 Value of Dny, when y = eax sin(bx + c)

1.2.6 Value of Dny, when y = eax cos(bx + c)

1.3 Leibnitz’s Theorem for the nth Derivative of the

Product of Two Functions

**Chapter 2 Partial Differentiation and Jacobians**

2.1 Introduction

2.2 Partial Derivatives

2.3 Partial Derivatives of Higher Order

2.4 Homogeneous Functions

2.5 Euler’s Theoremon Homogeneous Functions

2.6 Euler’s Theorem for Second Derivatives

2.7 Jacobians

2.8 Properties of Jacobians

2.8.1 If u and v are Functions of x and y, then (” ¶” (u,v))/(“¶” (x,y)) ´ (“¶” (x,y))/(“¶” (u,v)) = 1

2.8.2 If u and v are Functions of r and s which are Functions of

x and y, then( “¶” (u,v))/(“¶” (x,y))= (“¶” (u,v))/(“¶” (r,s)) ´ (“¶” (r,s))/(“¶” (u,v)) = 1

2.8.3 If u, v, w are Functionally Dependent Functions of Three

Independent Variables x, y, z then (“¶” (u,v,w))/(“¶” (x,y,z)) = 0

2.8.4 If the Transformations x = x(u, v) and y = y(u, v)

are made in the Double Integral òò f (x, y) dx dy, then

f (x, y) = f(u, v) and dx dy = |J| du dv, where J = (“¶” (x,y))/(“¶” (u,v))

**Chapter 3 Total Differentiation**

3.1 Introduction

3.2 Small Errors and Approximations

3.3 Differentiation of Implicit Functions

**Chapter 4 Maxima and Minima of Functions of Two Variables**

4.1 Introduction

4.2 Constrained Maxima and Minima

4.3 Lagrange’s Method of Undetermined Multipliers

**Chapter 5 Radius and Centre of Curvature **

5.1 Curvature and Radius of Curvature

5.2 Definition of Curvature

5.3 Some Basic Results

5.4 Formula for Radius of Curvature in Cartesian Co-ordinates

5.5 Formula for Radius of Curvature in Parametric Co-ordinates

5.6 Formula for Radius of Curvature in Polar Co-ordinates

5.7 Centre and Circle of Curvature

**Chapter 6 Evolutes and Envelopes**

6.1 Introduction

6.2 Method of Finding the Equation of the Envelope of a

Family of Curves

6.3 Evolute as the Envelope of Normals

## About The Author

**T. Veerarajan** is Dean (Retd.), Department of Mathematics, Velammal College of Engineering and Technology, Madurai, Tamil Nadu. A Gold Medalist from Madras University, he has had a brilliant academic career all through. He has 50 years of teaching experience at undergraduate and postgraduate levels in various established engineering colleges in Tamil Nadu including Anna University, Chennai.

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